AXISYMMETRIC VIBRATIONS OF ELASTIC BASES AND IDEAL LIQUID IN A RIGID ANNULAR CYLINDRICAL TANK - Kononov, Yu. M.; Dzhukha, Yu. O. (Computer Science and Applied Mathematics, 2016)
Interest in axisymmetric vibrations of elastic bases and liquid in an annular cylindrical tank is connected with the need to take into account vibrations of a liquid column between elastic bases. This is not necessary in the case of asymmetric vibrations of liquid in axisymmetric tank. Right annular cylindrical tank with elastic bases contains a heavy liquid. The bases are modeled by isotropic thin annular plates. It is considered three types of fixing of outer and inner contours: clamped, simply supported and free edge. The plates are under an action of tensile forces acting on a middle surface. It is assumed that the liquid is ideal, homogeneous and incompressible. The movement of the liquid is potential. The problem is considered in the linear formulation. Plates’ bending problem is represented as a sum of static and dynamic bending. Spectral problem is represented as a system of two inhomogeneous biharmonic equations. These equations govern coupled vibrations of plates and liquid. The general solution to each equation is the sum of the general solution of the related homogeneous equation and the particular solution. Mechanical parameters of the plate and the density of the liquid define the general solution of the homogeneous equation which contains an unknown frequency. Particular solution of the inhomogeneous equation takes the form of a series expansion in the eigenfunctions of ideal liquid vibrations in the annular cylindrical tank. From the condition of fixing of outer and inner contours of the plates, compatibility condition and condition of incompressibility of the liquid follows the frequency equation of natural joint axisymmetric vibrations of the elastic bases and the ideal liquid. In general this transcendental equation takes the form of a determinant of the ninth order. In general terms, the frequency spectrum of equation consists of two sets of frequencies corresponding to vibrations of upper and lower elastic bases. In the absence of gravity frequency equation is symmetrical referred to the upper and lower bases. On the basis of analytical and numerical investigations it may be concluded that: 1. The frequency spectrum consists of two sets of frequencies corresponding to vibrations of upper and lower elastic bases; 2. Under zero gravity axisymmetric vibrations are absent if the plates are massless or if one plate is absolutely rigid and another one is massless; 3. If the liquid has a free surface, then the frequency spectrum consists of two sets of frequencies corresponding to vibrations of the free surface and the elastic bottom. In a wide variation range of parameters of the mechanical system takes place a weak change of the first set of frequencies and a significant change of the second set of frequencies as well: 3.1. In most cases the first frequency of the second set depends on the dimensionless rigidity almost linearly; 3.2. The frequencies of the first set decrease slightly, and the frequencies of the second set decrease significantly when the depth of filling increase; 3.3. The frequencies of the second set in the case of the fixed outer and free inner contours and sufficiently small inner radius are closely equal to the corresponding frequencies in the case of a circular cylinder; 3.4. The series of the frequency equation converge fast enough. The time of finding solution of the frequency equation greatly goes up with an allowance for mass characteristics of the plates.